Problem: Simplify the following expression: $p = \dfrac{-45t^3 + 45t^2}{20t^3 + 15t^2}$ You can assume $t \neq 0$.
Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-45t^3 + 45t^2 = - (3\cdot3\cdot5 \cdot t \cdot t \cdot t) + (3\cdot3\cdot5 \cdot t \cdot t)$ The denominator can be factored: $20t^3 + 15t^2 = (2\cdot2\cdot5 \cdot t \cdot t \cdot t) + (3\cdot5 \cdot t \cdot t)$ The greatest common factor of all the terms is $5t^2$ Factoring out $5t^2$ gives us: $p = \dfrac{(5t^2)(-9t + 9)}{(5t^2)(4t + 3)}$ Dividing both the numerator and denominator by $5t^2$ gives: $p = \dfrac{-9t + 9}{4t + 3}$